Some Applications of Fractional Velocities

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Fractional Power ◽  
Leibniz Rule ◽  
Difference Quotient ◽  
Singular Behavior ◽  
Hölder Functions ◽  
Integer Order ◽  
Differentiable Functions ◽  
The Difference ◽  
Product Rules ◽  
Derivatives Of

AbstractFractional velocity is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. The fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. Relations to integer-order derivatives and other integral-based definitions are discussed.It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation.

COMMENTS ON "RIEMANN–CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]

Fractals ◽  
2015 ◽  
Vol 23 (02) ◽  
pp. 1575001 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Differential Geometry ◽  
Fractional Derivative ◽  
Power Function ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Space Time ◽  
Leibniz Rule ◽  
Integer Order ◽  
Fractal Space ◽  
Derivatives Of

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.

Energy Approximation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Main Lemma ◽  
The Other ◽  
Coarse Scale ◽  
Continuous Solutions ◽  
Material Derivative ◽  
Energy Variation ◽  
Energy Approximation ◽  
The Difference ◽  
Derivatives Of

This chapter presents the equations and calculations for energy approximation. It establishes the estimates (261) and (262) of the Main Lemma (10.1) for continuous solutions; these estimates state that we are able to accurately prescribe the energy that the correction adds to the solution, as well as bound the difference between the time derivatives of these two quantities. The chapter also introduces the proposition for prescribing energy, followed by the relevant computations. Each integral contributing to the other term can be estimated. Another proposition for estimating control over the rate of energy variation is given. Finally, the coarse scale material derivative is considered.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

Effects of avertin versus xylazine-ketamine anesthesia on cardiac function in normal mice

2001 ◽  
Vol 281 (5) ◽  
pp. H1938-H1945 ◽  
Author(s):  
Chari Y. T. Hart ◽  
John C. Burnett ◽  
Margaret M. Redfield
Keyword(s):  
Cardiac Function ◽  
Diastolic Pressure ◽  
Systolic Function ◽  
Pressure Rise ◽  
Left Ventricular ◽  
Lv Function ◽  
Ketamine Anesthesia ◽  
The Difference ◽  
And Function ◽  
Derivatives Of

Anesthetic regimens commonly administered during studies that assess cardiac structure and function in mice are xylazine-ketamine (XK) and avertin (AV). While it is known that XK anesthesia produces more bradycardia in the mouse, the effects of XK and AV on cardiac function have not been compared. We anesthetized normal adult male Swiss Webster mice with XK or AV. Transthoracic echocardiography and closed-chest cardiac catheterization were performed to assess heart rate (HR), left ventricular (LV) dimensions at end diastole and end systole (LVDd and LVDs, respectively), fractional shortening (FS), LV end-diastolic pressure (LVEDP), the time constant of isovolumic relaxation (τ), and the first derivatives of LV pressure rise and fall (dP/d t max and dP/d t min, respectively). During echocardiography, HR was lower in XK than AV mice (250 ± 14 beats/min in XK vs. 453 ± 24 beats/min in AV, P < 0.05). Preload was increased in XK mice (LVDd: 4.1 ± 0.08 mm in XK vs. 3.8 ± 0.09 mm in AV, P < 0.05). FS, a load-dependent index of systolic function, was increased in XK mice (45 ± 1.2% in XK vs. 40 ± 0.8% in AV, P < 0.05). At LV catheterization, the difference in HR with AV (453 ± 24 beats/min) and XK (342 ± 30 beats/min, P < 0.05) anesthesia was more variable, and no significant differences in systolic or diastolic function were seen in the group as a whole. However, in XK mice with HR <300 beats/min, LVEDP was increased (28 ± 5 vs. 6.2 ± 2 mmHg in mice with HR >300 beats/min, P < 0.05), whereas systolic (LV dP/d t max: 4,402 ± 798 vs. 8,250 ± 415 mmHg/s in mice with HR >300 beats/min, P < 0.05) and diastolic (τ: 23 ± 2 vs. 14 ± 1 ms in mice with HR >300 beats/min, P < 0.05) function were impaired. Compared with AV, XK produces profound bradycardia with effects on loading conditions and ventricular function. The disparate findings at echocardiography and LV catheterization underscore the importance of comprehensive assessment of LV function in the mouse.

Kinetic parameters of hexose transport in hybrids between malignant and nonmalignant cells

1983 ◽  
Vol 62 (1) ◽  
pp. 49-80
Author(s):  
M.K. White ◽  
M.E. Bramwell ◽  
H. Harris
Keyword(s):  
Kinetic Parameters ◽  
Malignant Cell ◽  
Hexose Transport ◽  
Hybrid Cells ◽  
Michaelis Constant ◽  
Independent Measurement ◽  
The Difference ◽  
Hexose Uptake ◽  
Derivatives Of

Matched pairs of isogeneic hybrid cells, in which one member of the pair was malignant and the other not, were used to examine the linkage between malignancy and functional alterations in hexose transport. The kinetic parameters of uptake of 2-deoxy-D-glucose were measured in a range of such hybrids, both human and murine. Some other malignant cell lines were also examined and were compared with non-tumorigenic derivatives of tumour cells selected by exposure to the lectin, wheat-germ agglutinin. In every case, malignancy, as defined by the ability of cells to grow progressively in vivo, was found to be linked to a decrease in the Michaelis constant of hexose uptake. Independent measurement of the transport and phosphorylation reactions involved in hexose uptake revealed that this decrease was determined by the membrane transport system. The difference in Michaelis constant between malignant and non-malignant cells was observed with 3-O-methylglucose, a hexose that is transported into the cell but not further metabolized. The activity of hexokinase in cell homogenates was higher than the level that would be required to cope with transport and showed no correlation with tumorigenicity. Measurement of the uptake of D-glucose itself, by a rapid filtration centrifugation method, gave results similar to those obtained with 2-deoxy-D-glucose.

scholarly journals A determinantal expression and a recursive relation of the Delannoy numbers

2021 ◽  
Vol 13 (2) ◽  
pp. 442-449 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Recursive Relation ◽  
Differentiable Functions ◽  
Delannoy Numbers ◽  
Derivatives Of ◽  
Determinantal Expression

Abstract In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

scholarly journals A Hopf-type Boundary Point Lemma for Pairs of Solutions to Quasilinear Equations

2019 ◽  
Vol 62 (3) ◽  
pp. 607-621 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Weak Solutions ◽  
Boundary Point ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Quasilinear Equations ◽  
Differentiable Functions ◽  
The Difference ◽  
Type Boundary

AbstractWe present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

scholarly journals Unusual Regularity in GC Retention of Simple Amino Acid Derivatives

2019 ◽  
Vol 6 (1) ◽  
pp. 3-14 ◽  
Author(s):  
Igor G. Zenkevich ◽  
Nino G. Todua ◽  
Anzor I. Mikaia
Keyword(s):  
Gas Chromatography ◽  
Retention Index ◽  
Methyl Esters ◽  
Elution Order ◽  
Alkyl Esters ◽  
Methyl Group ◽  
The Difference ◽  
Direct Use ◽  
Derivatives Of

Background: Application of simple regularities and general principles along with direct use of reference gas chromatography retention index data for reliable structure determination of compounds can be enhanced by determination of new regularities that are specific to certain structural elements. Objective: Revelation and interpretation of an anomaly in the elution order of alkyl esters of alkoxycarbonyl derivatives of glycine and alanine on standard and semi-standard non-polar phases. Method: Preliminary derivatization of amino acids to alkyl esters of N-alkoxycarbonyl analogs and interpretation of their gas chromatographic characteristics. Results: Alkyl esters of N-alkoxycarbonyl derivatives of alanine (Alkyl = C2H5, n- and iso-C3H7) elute prior to the same derivatives of glycine, despite the presence of an additional methyl group at C(2) in the molecule. Elution order is reversed for methyl esters of N-methoxycarbonyl derivatives. Conclusion: It is established that the peculiar behavior of alkyl esters of N-alkoxycarbonyl derivatives of glycine and alanine agrees with the concepts of gas chromatography and the known retention index regularities of organic compounds. A decrease of retention index values is a result of an introduction of an additional methyl group to a carbon atom connected to two polar fragments in a molecule like CH2XY. The dependence of the difference of retention index values for homologs of the types of CH3-CHXY and CH2XY vs. the total mass of fragments (X + Y) is similar to those for other sub-groups of analytes.

scholarly journals Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers

Materials ◽  
2019 ◽  
Vol 12 (17) ◽  
pp. 2725 ◽  
Author(s):  
Sergei Alexandrov ◽  
Lihui Lang ◽  
Elena Lyamina ◽  
Prashant P. Date
Keyword(s):  
Friction Surface ◽  
Yield Criterion ◽  
System Of Equations ◽  
Singular Behavior ◽  
Material Models ◽  
Metal Plasticity ◽  
Plane Strain Deformation ◽  
Stress Components ◽  
Approach Infinity ◽  
Derivatives Of

The present paper deals with plane strain deformation of incompressible polymers that obey quite a general pressure-dependent yield criterion. In general, the system of equations can be hyperbolic, parabolic, or elliptic. However, attention is concentrated on the hyperbolic regime and on the behavior of solutions near frictional interfaces, assuming that the regime of sliding occurs only if the friction surface coincides with an envelope of stress characteristics. The main reason for studying the behavior of solutions in the vicinity of envelopes of characteristics is that the solution cannot be extended beyond the envelope. This research is also motivated by available results in metal plasticity that the velocity field is singular near envelopes of characteristics (some space derivatives of velocity components approach infinity). In contrast to metal plasticity, it is shown that in the case of the material models adopted, all derivatives of velocity components are bounded but some derivatives of stress components approach infinity near the envelopes of stress characteristics. The exact asymptotic expansion of stress components is found. It is believed that this result is useful for developing numerical codes that should account for the singular behavior of the stress field.

Sign in / Sign up

Export Citation Format

Share Document